Theorem isgrpd2 13609

Description: Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2231, but we make an exception for theorems such as isgrpd2 13609 and ismndd 13525 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
isgrpd2.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isgrpd2.p	⊢ (𝜑 → + = (+g‘𝐺))
isgrpd2.z	⊢ (𝜑 → 0 = (0g‘𝐺))
isgrpd2.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
isgrpd2.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
isgrpd2.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpd2	⊢ (𝜑 → 𝐺 ∈ Grp)

Distinct variable groups:   𝑥, +   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥

Allowed substitution hints:   𝑁(𝑥)   0 (𝑥)

Proof of Theorem isgrpd2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isgrpd2.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		isgrpd2.p	. 2 ⊢ (𝜑 → + = (+g‘𝐺))
3		isgrpd2.z	. 2 ⊢ (𝜑 → 0 = (0g‘𝐺))
4		isgrpd2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
5		isgrpd2.n	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑁 ∈ 𝐵)
6		isgrpd2.j	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑁 + 𝑥) = 0 )
7		oveq1 6025	. . . . 5 ⊢ (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
8	7	eqeq1d 2240	. . . 4 ⊢ (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
9	8	rspcev 2910	. . 3 ⊢ ((𝑁 ∈ 𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
10	5, 6, 9	syl2anc 411	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
11	1, 2, 3, 4, 10	isgrpd2e 13608	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2202  ∃wrex 2511  ‘cfv 5326  (class class class)co 6018  Basecbs 13087  +_gcplusg 13165  0_gc0g 13344  Mndcmnd 13504  Grpcgrp 13588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021  df-grp 13591

This theorem is referenced by:  prdsgrpd  13697